Star Colouring of Regular Graphs Meets Weaving and Line Graphs

Abstract

For $q\in \mathbb {N}$, a $q$-star colouring of a graph $G$ is a proper $q$-colouring $f$ of $G$ such that there is no path $u,v,w,x$ in $G$ with $f(u)=f(w)$ and $f(v)=f(x)$ (the violating path need not be induced). For $p\ge 2$, Shalu and Antony (Discrete Math., 2022) proved that at least $p+2$ colours are required to star colour a $2p$-regular graph $G$, and characterised the class $\mathcal {G}$ of graphs $G$ for which $p+2$ colours suffices in terms of graph orientations. In the second author’s thesis (2023), we provided a characterisation of the class $\mathcal {G}$ in terms of locally constrained graph homomorphisms. In this paper, we characterise $\mathcal {G}$ in terms of weaving patterns of edge decompositions. We also show that the study of class $\mathcal {G}$ is tied to the theory of line graphs and line digraphs of complete graphs. We prove that if a $K_{1,p+1}$-free $2p$-regular graph $G$ with $p\ge 2$ is $(p+2)$-star colourable, then ${-2}$ and $p-2$ are eigenvalues of the adjacency matrix of $G$.